Halliday (1992: 23-5):
The result [of the "explosion" into language] is a semiotic of a new kind: a stratified, tristratal system in which meaning is 'twice cooked', thus incorporating a stratum of 'pure' content form. It is natural to represent this, as I have usually done myself, as 'meaning realised by wording, which is in turn realised by sound'. But it is also rather seriously misleading. If we follow Lemke's lead, interpreting language as a dynamic open system, we can arrive at a theoretically more accurate and more powerful account. Here the key concept is Lemke's principle of metaredundancy.
Consider a minimal semiotic system, such as a protolanguage — a system that is made up of simple signs. This is based on the principle of redundancy. When we say that contents p, q, r are "realised" respectively by expressions a, b, c, what this means is that there is a redundancy relation between them: given meaning p, we can predict sound or gesture a, and given sound or gesture a we can predict meaning p. This relationship is symmetrical; "redounds with" is equivalent both to "realises" and to "is realised by".
Let us now expand this into a non-minimal semiotic, one that is tri- rather than bi-stratal. The expressions a, b, c now realise WORDINGS l, m, n, while the wordings l, m, n realise MEANINGS p, q, r. In terms of redundancy, however, these are not two separate dyadic relationships. Rather, there is a METAredundancy such that p, q, r redounds not with l, m, n but with the redundancy of l, m, n with a, b, c; thus:
l, m, n ( a, b, c p, q, r ( (l, m, n ( a, b, c)
Why has it to be like this? Because there is not, in fact, a chain of dyadic relationships running through the system. (If there was, we would not need the extra stratum.) It is not the case, in other words, that p ( l and l ( a. p, q, r is realised by l, m, n; but the system at l, m, n is sorted out again for realisation by a, b, c, so that what p, q, r is actually realised by is the realisation of l, m, n by a, b, c. This is the fundamental distinction between redundancy and causality. If realisation was a causal relation, then it would chain: l is caused by a and p is caused by l - it would make no sense to say "p is caused by the causing of l by a". But realisation is not a causal relation; it is a redundancy relation, so that p redounds with the redundancy of l with a. To put it in more familiar terms, it is not that (i) meaning is realised by wording and wording is realised by sound, but that (ii) meaning is realised by the realisation of wording in sound.
We can of course reverse the direction, and say that sounding realises the realisation of meaning in wording:
p, q, r ( l, m, n (p, q, r ( l, m, n) ( a, b, c
For the purpose of phonological theory this is in fact the appropriate perspective, But for the purposes of construing the 'higher' levels, with language as connotative semiotic realising other semiotic systems of the culture, we need the first perspective. Thus when we extend 'upwards' to the context of situation, we can say that the context of situation s, t, u redounds with the redundancy of the discourse semantics p, q, r with the redundancy of the lexicogrammar l, m, n with the phonology a, b, c. Thus:
s, t , u ( (p, q, r, ( (l, m, n ( a, b, c))
Once the original protolinguistic redundancy has been transformed into metaredundancy in this way, the relation becomes an iterative one and so opens up the possibilities for construing, not only the context of situation, but also higher levels such as Hasan’s symbolic articulation and theme in verbal art, or Martin's strata of genre and ideology.
The metaredundancy notion thus formalises the stratal principle in semogenesis. What makes meaning indefinitely extendable is the evolutionary change from protolanguage to language — whereby instead of a simple plane with two interfaces to the material (the phenomenal), we have constructed a semiotic SPACE, a three-dimensional (potentially n-dimensional) system in which there is a purely symbolic mode of being between these two interfaces. It is this that we call grammar, or more explicitly lexicogrammar. Without this semiotic space, situated in the transduction from one purely symbolic mode to another, and hence not constrained by the need to interface directly with the phenomenal, we could not have a metafunctional organisation in the grammar, and we could not have the phenomenon of grammatical metaphor.
The metaredundancy theory explains the 'stratal' organisation of language, and the semiotic principle of realisation. It explains them synoptically: by treating realisation as a relation.